How would you know any thing at all that Jesus said had it not been written down??? How would your priests teach the gospel without books??? Or your Catechism??? Why did God have the OT prophets write down what he said???
So may I infer that you never wondered why the Great Commission did not say: “Go into all the world and hand out Bibles”?
I did once wonder about it. If that is the one needful thing, seems He would have been clear about that from the get go.
The "letters" were written in the 50's, James Peter and Paul were martyred in the mid 60's. Most scholars agree that Mark, the first Gospel was written some time around the year 70. Matthew and Luke were written independently of each other sometime either in the 80's or early 90's.
My guess would be that the Gospel (which literally means "Good News") was spread orally. Which if you think about it is still the way that most people hear it even today.